Asymptotic Analysis of Equivalences and Core-Structures in Kronecker-Style Graph Models

Abstract

Abstract-Growing interest in modeling large, complex networks has spurred significant research into generative graph models. These models not only allow users to gain insight into the processes underlying networks, but also provide synthetic data which allows algorithm scalability testing and addresses privacy concerns. Kronecker-style models (e.g. SKG and R-MAT) are often used due to their scalability and ability to mimic key properties of real-world networks (e.g. diameter and degree distribution). Although a few papers theoretically establish these models' behavior for specific parameters, many claims used to justify the use of these models in various applications are supported only by empirical evaluations. In this work, we prove several results using asymptotic analysis which illustrate that empirical studies may not fully capture the true behavior of the models.Paramount to the widespread adoption of Kronecker-style models was the introduction of a linear-time edge-sampling variant (R-MAT), which existing literature typically treats as interchangeable with SKG. We prove that although several R-MAT formulations are asymptotically equivalent, their behavior diverges from that of SKG. Further, we show these results are experimentally observable even at relatively small graph sizes. Second, we consider a case where asymptotic analysis reveals unexpected behavior within a given model. One of the criticisms of using Kronecker-style models has been that they are unable to generate the deep core-structures commonly observed in real-world data. We prove that in fact, for some parameter values, all the Kronecker-style models generate graphs whose maximum core depth grows as a function of the size of the network-including in the region of the parameter space most commonly used in prior work. Our results also illustrate why this behavior may be difficult to observe for moderate graph sizes, and highlight the dangers of extrapolating model-wide claims from empirical results.